Optimal constants of smoothing estimates for the Dirac equation in arbitrary dimensions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913637479743488 |
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| author | Suzuki, Soichiro |
| author_facet | Suzuki, Soichiro |
| contents | We give optimal constants of smoothing estimates for the $d$-dimensional free Dirac equation for any $d \geq 2$. Our main abstract theorem shows that the optimal constant $C$ of smoothing estimate associated with a spatial weight $w$ and smoothing function $ψ$ is given by $(2π)^{d-1} C = \sup_{k \in \mathbb{N}} \sup_{r > 0} \widetildeλ_k(r)$, where $\{ \widetildeλ_k \}$ is a certain sequence of functions defined via integral formulae involving $(w, ψ)$. This is an analogue of a similar result for Schrödinger equations given by Bez--Saito--Sugimoto (2015), and also extends previous results of Ikoma (2022) and Ikoma--Suzuki (2024) for $d=2, 3$ to any dimensions $d \geq 2$. In order to prove this, we establish a modified version of the spherical harmonics decomposition of $L^2(\mathbb{S}^{d-1})$, which suits well with the Dirac operator and allows us to find optimal constants. Furthermore, using our abstract theorem, we give explicit values of optimal constants associated with typical examples of $(w, ψ)$. As it turns out, optimal constants for Dirac equations can be written explicitly in many cases, even in the cases that it is impossible for Schrödinger equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_00949 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal constants of smoothing estimates for the Dirac equation in arbitrary dimensions Suzuki, Soichiro Analysis of PDEs Classical Analysis and ODEs 33C55, 35B65, 35Q41, 42B10 We give optimal constants of smoothing estimates for the $d$-dimensional free Dirac equation for any $d \geq 2$. Our main abstract theorem shows that the optimal constant $C$ of smoothing estimate associated with a spatial weight $w$ and smoothing function $ψ$ is given by $(2π)^{d-1} C = \sup_{k \in \mathbb{N}} \sup_{r > 0} \widetildeλ_k(r)$, where $\{ \widetildeλ_k \}$ is a certain sequence of functions defined via integral formulae involving $(w, ψ)$. This is an analogue of a similar result for Schrödinger equations given by Bez--Saito--Sugimoto (2015), and also extends previous results of Ikoma (2022) and Ikoma--Suzuki (2024) for $d=2, 3$ to any dimensions $d \geq 2$. In order to prove this, we establish a modified version of the spherical harmonics decomposition of $L^2(\mathbb{S}^{d-1})$, which suits well with the Dirac operator and allows us to find optimal constants. Furthermore, using our abstract theorem, we give explicit values of optimal constants associated with typical examples of $(w, ψ)$. As it turns out, optimal constants for Dirac equations can be written explicitly in many cases, even in the cases that it is impossible for Schrödinger equations. |
| title | Optimal constants of smoothing estimates for the Dirac equation in arbitrary dimensions |
| topic | Analysis of PDEs Classical Analysis and ODEs 33C55, 35B65, 35Q41, 42B10 |
| url | https://arxiv.org/abs/2501.00949 |